Multi-scale Deep Neural Networks for Solving High Dimensional PDEs

TL;DR

This paper introduces a multi-scale deep neural network architecture that enhances the approximation of high-frequency, high-dimensional functions and accelerates solutions to high-dimensional PDEs, validated through numerical experiments.

Contribution

The paper proposes a novel multi-scale DNN with radial scaling and compact support activation functions, improving high-frequency and high-dimensional function approximation capabilities.

Findings

Enhanced ability to capture high-frequency components.

Faster convergence in solving high-dimensional PDEs.

Validated effectiveness through numerical experiments.

Abstract

In this paper, we propose the idea of radial scaling in frequency domain and activation functions with compact support to produce a multi-scale DNN (MscaleDNN), which will have the multi-scale capability in approximating high frequency and high dimensional functions and speeding up the solution of high dimensional PDEs. Numerical results on high dimensional function fitting and solutions of high dimensional PDEs, using loss functions with either Ritz energy or least squared PDE residuals, have validated the increased power of multi-scale resolution and high frequency capturing of the proposed MscaleDNN.